Categorifying the Knizhnik-Zamolodchikov Connection

TL;DR

This paper constructs a categorified version of the Knizhnik-Zamolodchikov connection within higher gauge theory, introducing a flat 2-connection and differential crossed modules to advance the mathematical framework.

Contribution

It introduces a categorification of the Knizhnik-Zamolodchikov connection using differential crossed modules and develops the representation theory for these structures.

Findings

Defined the differential crossed module of horizontal 2-chord diagrams.

Categorified the 4-term relation in higher gauge theory.

Formulated the notion of an infinitesimal 2-R matrix.

Abstract

In the context of higher gauge theory, we construct a flat and fake flat 2-connection, in the configuration space of $n$ particles in the complex plane, categorifying the Knizhnik-Zamolodchikov connection. To this end, we define the differential crossed module of horizontal 2-chord diagrams, categorifying the Lie algebra of horizontal chord diagrams in a set of $n$ parallel copies of the interval. This therefore yields a categorification of the 4-term relation. We carefully discuss the representation theory of differential crossed modules in chain-complexes of vector spaces, which makes it possible to formulate the notion of an infinitesimal 2-R matrix in a differential crossed module.